We present and implement a parquet approximation within the dual-fermion formalism based on a partial bosonization of the dual vertex function which substantially reduces the computational cost of the calculation. The method relies on splitting the vertex exactly into single-boson exchange contributions and a residual four-fermion vertex, which physically embody, respectively, long- and short-range spatial correlations. After recasting the parquet equations in terms of the residual vertex, these are solved using the truncated-unity method of Eckhardt et al. [Phys. Rev. B 101, 155104 (2020)2469-995010.1103/PhysRevB.101.155104], which allows for a rapid convergence with the number of form factors in different regimes. While our numerical treatment of the parquet equations can be restricted to only a few Matsubara frequencies, reminiscent of Astretsov et al. [Phys. Rev. B 101, 075109 (2020)2469-995010.1103/PhysRevB.101.075109], the one- and two-particle spectral information is fully retained. In applications to the two-dimensional Hubbard model the method agrees quantitatively with a stochastic summation of diagrams over a wide range of parameters.